109 lines
4.7 KiB
Markdown
109 lines
4.7 KiB
Markdown
---
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title: Units, dimensions and conversion
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tags: units, dimensions, si, conversion
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summary: How physical quantities carry their unit, why a unit is a type rather than a label, and the conventions that make an arithmetic answer meaningful.
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---
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# Units, dimensions and conversion
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A number without a unit is not a physical quantity. "3" might be three metres,
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three seconds, or three kilometres per hour, and the three do not describe the
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same thing. The purpose of a unit system is to make an arithmetic result
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interpretable, which means the unit is not a note attached to a value but part
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of the value itself.
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## Definition
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A **quantity** is a magnitude together with a **unit**. The **dimension** is
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the kind of thing measured, such as length, time, or mass; a unit is a
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particular choice of magnitude within that dimension.
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The base dimensions of the International System of Units, and the units chosen
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for them, are:
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| Dimension | Unit | Defined by |
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|-----------|------|-----------|
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| Length | metre | Speed of light in vacuum |
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| Mass | kilogram | Planck constant |
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| Time | second | Cesium hyperfine transition |
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| Electric current | ampere | Elementary charge |
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| Thermodynamic temperature | kelvin | Boltzmann constant |
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| Amount of substance | mole | Avogadro constant |
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| Luminous intensity | candela | Defined by the other six |
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Everything else is **derived**: newton from kilogram, metre and second; joule
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from newton and metre; watt from joule and second.
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## Mathematics
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A unit is a conversion factor to a base unit. Length may be expressed in
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metres, kilometres or feet; each is a number multiplied by a factor to give
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metres. Dimensionally aware arithmetic keeps track of that factor, so
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$$
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1\,\text{km} + 500\,\text{m} = 1500\,\text{m}, \quad \text{never } 1 + 500 = 501
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$$
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The rule is that the conversion factor travels with the value. A quantity of
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type length carries the factor to metres with it, so adding it to another
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length applies both factors at once.
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A dimension behaves like a type in a type system. Adding a length to a time is
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a dimension error, in the same way that adding a string to an integer would be
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one in a compiler that tracked those types.
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### Scaling laws
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Dimension reasoning earns its keep in scaling arguments, where an error shows
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up as an impossible exponent. The gravitational perturbation at distance $r$
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from a body of mass $M$ must scale as
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$$
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a \propto \frac{G M}{r^2}
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$$
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because $G$ has units of $\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$; any other combination either does
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not have the dimensions of an acceleration or does have them only for a
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particular case, which is the signal that the argument is incomplete.
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## Unit conventions
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- **Angles.** An angle is dimensionless in principle, because a full turn is
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one. It is written in radians in most formulae, and in degrees, arcminutes
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and arcseconds where it is read by a person. The relationship is
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$180 / \pi$ degrees per radian. A rotation described in axis and angle uses
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the angle in a plane and the axis through the pole; the two conventions
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differ, which is why a rotation is usually written explicitly as
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right-hand or left-hand.
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- **Prefixes.** SI prefixes are powers of ten: kilo is $10^3$, mega $10^6$,
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micro $10^{-6}$. A kilometre is a thousand metres, never a thousand miles.
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- **Binary prefixes.** A mebibyte is $2^{20}$ bytes, not $10^6$. The distinction
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matters wherever a size is compared against a capacity.
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- **Units in the field.** Orbital work is written in kilometres and seconds,
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not metres, because the quantities are large and the models are fitted to
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those units. A propagation model expressed in metres and a catalogue
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expressed in kilometres will disagree by a factor of a thousand, silently.
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## Applicability
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Unit arithmetic prevents a class of error that is otherwise easy to make and
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hard to see, because the numbers stay plausible. It does not prevent a
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different class: the same unit used with two different conventions. A height
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above the equatorial radius and a height above the ellipsoid differ by tens of
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kilometres; both are lengths, and neither is wrong.
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The measure a quantity is taken against matters as much as its unit. An epoch
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expressed in seconds from an unspecified origin is not comparable with one
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expressed in a Modified Julian Date, even though both are times.
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## Standards
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- SI: *The International System of Units (SI)*, BIPM, most recent edition
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- Non-SI units accepted for use with SI: the same document, section 4
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- Earth reference models: NIMA TR8350.2 for WGS 84
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## See also
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- [[numerical-methods]] for what the numbers are computed with
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- [[time-scales]] for a quantity whose unit hides an entire convention
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- [[geodesy-wgs84]] for the two measures a height is taken against |